Electron scattering states at solid surfaces calculated with realistic potentials

Scattering states with LEED asymptotics are calculated for a general non-muffin tin potential, as e.g. for a pseudopotential with a suitable barrier and image potential part. The latter applies especially to the case of low lying conduction bands. The wave function is described with a reciprocal lattice representation parallel to the surface and a discretization of the real space perpendicular to the surface. The Schroedinger equation leads to a system of linear one-dimensional equations. The asymptotic boundary value problem is confined via the quantum transmitting boundary method to a finite interval. The solutions are obtained basing on a multigrid technique which yields a fast and reliable algorithm. The influence of the boundary conditions, the accuracy and the rate of convergence with several solvers are discussed. The resulting charge densities are investigated.Comment: 5 pages, 4 figures, copyright and acknowledgment added, typos etc. correcte

Geometric Structures in Tensor Representations (Final Release)

The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce, in the underlying algebraic tensor space, the set of tensors in a tree-based format with either bounded or fixed tree-based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the Tensor Train format). In particular, we show that the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank of an algebraic tensor product of normed vector spaces is an analytic Banach manifold. Indeed, the manifold geometry for the set of tensors with fixed tree-based rank is induced by a fibre bundle structure and the manifold geometry for the set of tensors with bounded tree-based rank is given by a finite union of connected components. In order to describe the relationship between these manifolds and the natural ambient space, we introduce the definition of topological tensor spaces in the tree-based format. We prove under natural conditions that any tensor of the topological tensor space under consideration admits best approximations in the manifold of tensors in the tree-based format with bounded tree-based rank. In this framework, we also show that the tangent (Banach) space at a given tensor is a complemented subspace in the natural ambient tensor Banach space and hence the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank is an immersed submanifold. This fact allows us to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces.Comment: Some errors are corrected and Lemma 3.22 is improve

On the Convergence of Alternating Least Squares Optimisation in Tensor Format Representations

The approximation of tensors is important for the efficient numerical treatment of high dimensional problems, but it remains an extremely challenging task. One of the most popular approach to tensor approximation is the alternating least squares method. In our study, the convergence of the alternating least squares algorithm is considered. The analysis is done for arbitrary tensor format representations and based on the multiliearity of the tensor format. In tensor format representation techniques, tensors are approximated by multilinear combinations of objects lower dimensionality. The resulting reduction of dimensionality not only reduces the amount of required storage but also the computational effort.Comment: arXiv admin note: text overlap with arXiv:1503.0543

SYNTHETIC METHODS FOR ESTER BOND FORMATION AND CONFORMATIONAL ANALYSIS OF ESTER-CONTAINING CARBOHYDRATES

This dissertation encompasses work related to synthetic methods for the formation of ester linkages in organic compounds, as well as the investigation of the conformational influence of the ester functional group on the flexibility of inter-saccharide linkages, specifically, and the solution phase structure of ester-containing carbohydrate derivatives, in general. Stereoselective reactions are an important part of the field of asymmetric synthesis and an understanding of their underlying mechanistic principles is essential for rational method development. Here, the exploration of a diastereoselective O-acylation reaction on a trans-2-substituted cyclohexanol scaffold is presented, along with possible reasons for the observed reversal of stereoselectivity dependent on the presence or absence of an achiral amine catalyst. In particular, this work establishes a structureactivity relationship with regard to the trans-2-substituent and its role as a chiral auxiliary in the reversal of diastereoselectivity. In the second part, the synthesis of various ester-linked carbohydrate derivatives, and their conformational analysis is presented. Using multidimensional NMR experiments and computational methods, the compounds’ solution-phase structures were established and the effect of the ester functional group on the molecules’ flexibility and three-dimensional (3D) structure was investigated and compared to ether or glycosidic linkages. To aid in this, a novel Karplus equation for the C(sp2)OCH angle in esterlinked carbohydrates was developed on the basis of a model ester-linked carbohydrate. This equation describes the sinusoidal relationship between the C(sp2)OCH dihedral angle and the corresponding 3JCH coupling constant that can be determined from a JHMBC NMR experiment. The insights from this research will be useful in describing the 3D structure of naturally occurring and lab-made ester-linked derivatives of carbohydrates, as well as guiding the de novo-design of carbohydrate based compounds with specific shape constraints for its use as enzyme inhibitors or similar targets. In addition, the above project led to the development of a methodology for the synthesis of symmetrical ester molecules from primary alcohols using a mild oxidative esterification reaction, which proceeds in hydrous solvents using a nitrosyl radical catalyst. The reaction could be performed with a variety of alcohols and the resulting compounds are of interest in the fragrance and flavor industries

Hierarchical matrix techniques for low- and high-frequency Helmholtz problems

In this paper, we discuss the application of hierarchical matrix techniques to the solution of Helmholtz problems with large wave number κ in 2D. We consider the Brakhage-Werner integral formulation of the problem discretized by the Galerkin boundary-element method. The dense n × n Galerkin matrix arising from this approach is represented by a sum of an -matrix and an 2-matrix, two different hierarchical matrix formats. A well-known multipole expansion is used to construct the 2-matrix. We present a new approach to dealing with the numerical instability problems of this expansion: the parts of the matrix that can cause problems are approximated in a stable way by an -matrix. Algebraic recompression methods are used to reduce the storage and the complexity of arithmetical operations of the -matrix. Further, an approximate LU decomposition of such a recompressed -matrix is an effective preconditioner. We prove that the construction of the matrices as well as the matrix-vector product can be performed in almost linear time in the number of unknowns. Numerical experiments for scattering problems in 2D are presented, where the linear systems are solved by a preconditioned iterative metho

A new finite element approach for problems containing small geometric details

summary:In this paper a new finite element approach is presented which allows the discretization of PDEs on domains containing small micro-structures with extremely few degrees of freedom. The applications of these so-called Composite Finite Elements are two-fold. They allow the efficient use of multi-grid methods to problems on complicated domains where, otherwise, it is not possible to obtain very coarse discretizations with standard finite elements. Furthermore, they provide a tool for discrete homogenization of PDEs without requiring periodicity of the data

On the interconnection between the higher-order singular values of real tensors

A higher-order tensor allows several possible matricizations (reshapes into matrices). The simultaneous decay of singular values of such matricizations has crucial implications on the low-rank approximability of the tensor via higher-order singular value decomposition. It is therefore an interesting question which simultaneous properties the singular values of different tensor matricizations actually can have, but it has not received the deserved attention so far. In this paper, preliminary investigations in this direction are conducted. While it is clear that the singular values in different matricizations cannot be prescribed completely independent from each other, numerical experiments suggest that sufficiently small, but otherwise arbitrary perturbations preserve feasibility. An alternating projection heuristic is proposed for constructing tensors with prescribed singular values (assuming their feasibility). Regarding the related problem of characterising sets of tensors having the same singular values in specified matricizations, it is noted that orthogonal equivalence under multilinear matrix multiplication is a sufficient condition for two tensors to have the same singular values in all principal, Tucker-type matricizations, but, in contrast to the matrix case, not necessary. An explicit example of this phenomenon is given